Transactional e-mail: How to template 'sales_email_order_items' (or how to override adminhtml template)

Actually it refers to a Layout XML handle you can find in the sales.xml file around line 268.

There you can find the following tags

<sales_email_order_items>
    <block type="sales/order_email_items" name="items" template="email/order/items.phtml">
        <action method="addItemRender"><type>default</type><block>sales/order_email_items_order_default</block><template>email/order/items/order/default.phtml</template></action>
        <action method="addItemRender"><type>grouped</type><block>sales/order_email_items_order_grouped</block><template>email/order/items/order/default.phtml</template></action>
        <block type="sales/order_totals" name="order_totals" template="sales/order/totals.phtml">
            <action method="setLabelProperties"><value>colspan="3" align="right" style="padding:3px 9px"</value></action>
            <action method="setValueProperties"><value>align="right" style="padding:3px 9px"</value></action>
            <block type="tax/sales_order_tax" name="tax" template="tax/order/tax.phtml">
                <action method="setIsPlaneMode"><value>1</value></action>
            </block>
        </block>
    </block>
    <block type="core/text_list" name="additional.product.info" />
</sales_email_order_items>

You can copy this to your own themes local.xml and edit whatever you need. In your case it would be resetting the template file like so:-

<sales_email_order_items>
    <reference name="items">
        <action method="setTemplate">
            <template>yourdirectory/order/items.phtml</template>
        </action>
    </reference>
</sales_email_order_items>

You can see which template "sales_email_order_items" refers when looking into

app/design/frontend/base/default/layout/sales.xml
app/design/frontend/base/default/layout/bundle.xml
app/design/frontend/base/default/layout/downloadable.xml

In app/design/frontend/base/default/layout/sales.xml for example you will see:

<sales_email_order_items>
        <block type="sales/order_email_items" name="items" template="email/order/items.phtml">
            <action method="addItemRender"><type>default</type><block>sales/order_email_items_order_default</block><template>email/order/items/order/default.phtml</template></action>
            <action method="addItemRender"><type>grouped</type><block>sales/order_email_items_order_grouped</block><template>email/order/items/order/default.phtml</template></action>
            <block type="sales/order_totals" name="order_totals" template="sales/order/totals.phtml">
                <action method="setLabelProperties"><value>colspan="3" align="right" style="padding:3px 9px"</value></action>
                <action method="setValueProperties"><value>align="right" style="padding:3px 9px"</value></action>
                <block type="tax/sales_order_tax" name="tax" template="tax/order/tax.phtml">
                    <action method="setIsPlaneMode"><value>1</value></action>
                </block>
            </block>
        </block>
        <block type="core/text_list" name="additional.product.info" />
    </sales_email_order_items>

Here you can find out the path: template="email/order/items.phtml"

Your directory structure /app/design/our_theme/default/default/template/email/order/items.phtml seems wrong, especially the part /app/design/our_theme/default/default/template... - it should be /app/design/our_theme/default/template... (one default less, i guess).